The Yamabe problem on CR manifolds

Jerison, David; Lee, John M.

doi:10.4310/jdg/1214440849

Cited by 282 publications

(328 citation statements)

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“…Indeed, the Cayley transform Ψ c is a biholomorphism between the unit ball in C n+1 and the Siegel domain; when restricted to the respective boundaries it gives a CR equivalence between S 2m−1 minus a point and H n ≃ ∂Ω n+1 (which is the CR equivalent of the stereographic projection of the unit sphere on the Euclidean space), see also section 3 and [48]. Combining Theorem 1.2 with the fractional Sobolev embedding on the Heisenberg group from [26] we obtain the following energy inequality:…”

Section: Equality Is Attained If and Only Ifmentioning

confidence: 98%

“…Indeed, ifφ = v 2 m−γ ϕ is another defining function for M and v| M = w, which gives a relation between the contact forms aŝ θ = w 2 m−γ θ, then the corresponding operator is given by Pθ γ (·) = w − m+γ m−γ P θ γ (w ·). In particular, for γ = 1, we obtain the CR Yamabe operator of Jerison-Lee [48] …”

Section: Scattering Theory and The Conformal Fractional Sub-laplacianmentioning

confidence: 99%

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An extension problem for the CR fractional Laplacian

Frank

González

Monticelli

et al. 2015

Advances in Mathematics

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Section: Equality Is Attained If and Only Ifmentioning

confidence: 98%

Section: Scattering Theory and The Conformal Fractional Sub-laplacianmentioning

confidence: 99%

An extension problem for the CR fractional Laplacian

Frank

González

Monticelli

et al. 2015

Advances in Mathematics

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“…For more details and explanations, we refer to [56]. Let U be a relatively compact open subset of a normal coordinate neighborhood ζ , as in Theorem 3.43.…”

Section: Definition 341mentioning

confidence: 99%

“…Our aim is to find suitable conditions on K which enable to prove the existence on M of a contact formθ C R conformal to θ , having the function K as Webster scalar curvature, Rθ = K . We writeθ = u 2 n θ, where u is a positive function defined on M. We obtain the following transformation law for the conformal Laplacians −L θ and −L θ , see [56].…”

Section: Preliminariesmentioning

confidence: 99%

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Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Gamara

2017

Arab. J. Math.

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Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings.Mathematics Subject Classification 58E05 · 57R70 · 53C21 · 53C17

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Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂⁿ

Cohn

2004

Comm Pure Appl Math

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The main results of this paper concern sharp constants for the Moser-Trudinger inequalities on spheres in complex space C n . We derive Moser-Trudinger inequalities for smooth functions and holomorphic functions with different sharp constants (see Theorem 1.1). The sharp Moser-Trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the CR setting. Though there is a close connection in spirit between inequalities proven here on complex spheres and those on the Heisenberg group for functions with compact support in any finite domain proven earlier by the same authors [17], derivation of the sharp constants for Moser-Trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the Heisenberg group. Variants of Moser-Onofri-type inequalities are also given on complex spheres as applications of our sharp inequalities (see Theorems 1.2 and 1.3). One of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients (see Theorem 1.4).

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The Yamabe problem on CR manifolds

Cited by 282 publications

References 17 publications

An extension problem for the CR fractional Laplacian

An extension problem for the CR fractional Laplacian

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂⁿ

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The Yamabe problem on CR manifolds

Cited by 282 publications

References 17 publications

An extension problem for the CR fractional Laplacian

An extension problem for the CR fractional Laplacian

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂn

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Sharp constants for Moser‐Trudinger inequalities on spheres in complex space ℂⁿ